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Factorization of Flow Profile Data in Production and Injection Wells Based on Bayesian Modeling*

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Factorization of Flow Profile Data in Production and Injection Wells Based on Bayesian Modeling* Factorization of flow profile data in production and injection wells based on Bayesian modeling* Konstantin Sidelnikov1 and Rinat Faizullin2 1 Department of Reservoir Energy State Control, Izhevsk Petroleum Scientific Center, Izhevsk, Russia 2 Department of Management, Kalashnikov Izhevsk State Technical University, Izhevsk, Rus- sia [email protected] Abstract. This problem is ill-posed, and the process of adding information in order to solve it is needed (regularization). Several methods of deterministic regularization based on 2 -norm minimization of the unknown vector are con- sidered. It is shown that such approaches do not take into account the petro- physical properties of the reservoir and cannot cover all possible factorization combinations. Bayesian regularization is proposed to factorize the flow profile data. According to this method, all relative factors are defined by the corre- sponding probability distribution functions. Core studies are used to determine the joint probability distribution of rock permeability and porosity. Layer pro- ductivity ratio distributions are calculated separately for each well based on its log interpretation data. Bayesian statistical inference is used to obtain the gen- eral drawdown ratio distribution for the entire field. This approach was tested on real data obtained from three fields. Keywords: Regularization, Bayes’ theorem, statistical inference, well test, flow profile, productivity index, injectivity index 1 Introduction In the process of analyzing the oil field development, it is required to solve the prob- lems of determining the injected water front, assessing the oil recovery factor sepa- rately for the producing formations, etc. In order to distribute the cumulative oil pro- duction and the volume of injected water between the producing formations, it is nec- essary to have the values of the oil and water flow rate for each formation separately [1-7]. Investigation of the flow profiles in formations allows obtaining the distribution of the produced and injected fluid over the entire cross-section of the pay zone. As a result, the dependence of the amount of produced or injected fluids on the depth of the completed interval is established. To obtain flow profiles over the formation thick- * Copyright © 2021 for this paper by its authors. Use permitted under Creative Commons License Attribu- tion 4.0 International (CC BY 4.0). ness, the results of measurements by flow meters are usually used, as well as ther- mometry data in the wellbore [8-15]. Bayesian methods are widely used in the practice of developing hydrocarbon fields: processing the results of well tests [5-6; 11] and well logging [9; 12-13], in reservoir simulation [7-8; 10], in statistical prediction of oil-field performance [3-4]. 2 Materials and methods Formulation of the problem. The paper discusses the results of determining the flow profile for a two-layered res- ervoir (Figure 1). It is also assumed that the inflow data from the investigated well interval contain the value of the total oil and water production rate without exact in- formation about fractional flow of each fluid. Fig. 1. Well inflow model. Thus, as a result of the production log, the following flow rate ratio for 2 produc- tion layers becomes known: Q q 1 , (1) Q2 Where: Qi JPP ii wf JPi ii , 1, 2 (2) Ji – productivity of i -th layer; Pi – drawdown for the i-th layer in the case of production ("+" sign) or injection ("–" sign), respectively. The problem is to factorize (1) resulted in decomposition of the known q flow rate ratio into the product of the following unknown ratios: q j p (3) Where j JJ1 2 – productivity / injectivity ratio; p PP1 2 – drawdown ratio. Possible interpretations of nonuniform recovery of reserves based on factorization (3) are: ─ For relatively homogeneous reservoirs in terms of productivity, it can be approxi- mately assumed that j 1 ; therefore, the observed flow profile will be determined mainly by the ratio p, which will depend on the initial reserve and dynamics of the reservoir energy consumption in relation to each reservoir; ─ In the case of hydrodynamic equilibrium between the reservoirs and sufficiently fast process of repressuring between them, it can be assumed that p 1; therefore the main factor of the flow profile non-uniformity will be the heterogeneity of the reservoir properties, expressed as a ratio j. There may also be other possible values of the productivity and drawdown ratios, reflecting more complex hydrodynamic processes of multilayer reservoir develop- ment. Obviously, under such conditions, factorization problem (3) has no solution with- out additional information. In particular, if it is possible to directly calculate the pro- ductivity Ji of each interval, then with the known q, it is possible to obtain the draw- down for each reservoir. However, direct calculation of Ji requires knowing the transmissibility of each layer, the current inflow regime of the well, the reservoir ge- ometry, etc. [15]. The absence or incompleteness of such information generally pre- cludes this approach. It is known that, to control the development of oil and gas fields, in addition to the production log, well testing is carried out as well. The values of the productivity index and reservoir pressure obtained from the results of these studies, as a rule, are integral in nature and determine the properties of the multilayered system only as a whole. Thus, on the basis of well testing, only the total fluid flow rate (injection) can be fac- tored: QJ P (4) Where Q Q1 Q 2 , J J1 J 2 and weighted average drawdown: J J P1 P 2 P (5) J1 J 2 Obviously, however, for an unambiguous factorization (3) one well-known expres- sion in the form (4) is not enough. Moreover, this problem can be attributed to an ill- posed problem, the solution to which requires some additional constraints to its condi- tions (regularization) [16]. Deterministic regularization. When solving factorization problem (3) on the basis of (4), in fact, there is a vector of four unknowns: T u JJ1 2 P 1 P 2 (6) And a system of three equations: Q1 J 1 P 1 Q2 J 2 P 2 (7) J J1 J 2 The system of equations (7) has many solutions; therefore, one of the methods for obtaining a unique solution is to impose an additional constraint on the norm of the vector space [16]. For example, consider the minimization of the 2 -norm of the vector (6): 2 u uuT min (8) 2 Subject to the fulfillment of (7). Problem (8) formally belongs to the class of nonlinear programming problems [17] due to additional conditions (7), some of which are nonlinear functions with respect to variables. Therefore, a special solution method is required, suitable for constrained optimization problems [17]. Conditions (7) can be linearized by replacing the variables for drawdown as fol- lows: T x JJ1 21 P 1 1 P 2 (9) As a result, system (7) can be represented as: Ax b (10) Where: 1 1 0 0 J A1 0Q 0 , b 0 1 (11) 0 1 0Q2 0 Further, there are two essentially equivalent ways of solving the problem: x2 xT x min 2 (12) Method 1 (optimization methods). On the one hand, problem (12) with constraints (11) is a quadratic programming problem [17]. Due to the fact that all constraints are equalities (there are no inequalities), it can be reduced to solving a system of linear equations: I AT x 0 (13) A 0 λ b Where I is the identity matrix, the elements of the main diagonal of which are equal to one; λ is the vector of Lagrange multipliers that appear along with the solu- tion x. Method 2 (methods of linear algebra). On the other hand, system (10) itself be- longs to the class of underdetermined systems of linear equations; therefore, one of the ways to solve it is to obtain a system of normal equations, in which the number of equations will already be equal to the number of unknowns: AAxT Ab T (14) Unfortunately, for a given matrix A in the form (11), system (14) has no solution, since the determinant of a normal matrix ATA is 0. Nevertheless, solution (10) remains possible on the basis of special algorithms for decomposition of rectangular matrices: QR decomposition, singular value decomposi- tion (SVD), etc. Thus, factorization problem (3) can be solved in the formulation of minimizing 2 - norm of either vector u (8) or vector x (12). Table 1 shows the results of calculations in two ways. Here Q = 50 m3 / day and J = 1 m3 / day / bar. As follows from the results in table 1, the solutions obtained in two ways differ significantly from each other, with the exception of the case q = 1. This is due, of course, to the fact that in one case the values of the drawdown are minimized, and in the other case, their reciprocal values. Table 1. Solutions based on minimizing the 2 -norm. Solution in the form u Solution in the form x q J1 , J 2 , P1 , P2 , J1 , J 2 , P1 , P2 , m3/day/bar m3/day/bar bar bar m3/day/bar m3/day/bar bar bar 50/50 0.500 0.500 50.0 50.0 0.500 0.500 50.0 50.0 70/30 0.637 0.363 54.9 41.4 0.501 0.499 69.9 30.1 90/10 0.812 0.188 55.4 26.6 0.510 0.490 88.3 10.2 99/1 0.955 0.045 51.8 11.2 0.833 0.167 59.4 3.0 It is also interesting to note that for the most frequently encountered in practice range of 1q 9 values, the solution in the form of x corresponds to formations that are practically homogeneous in terms of productivity.
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